Largest Even

Why do this problem?

This problem encourages children to work together to develop a method for finding a solution which will always work. It will also help to reinforce understanding of odd and even numbers.

Possible approach

You could introduce the challenge using two sets of 0-9 digit cards. Hold up a 3 and ask the class which digit they could choose to put with the 3 to make a two-digit even number. Take a few suggestions and encourage learners to find all the possibilities (30, 32, 34, 36, 38), listing them on the board as you go. How do they know that each of these is even? What do they notice about these numbers? Encourage the class to explain why the 3 must go in the tens column if we are making an even number.

Repeat this, perhaps by holding up an even digit, for example 6, this time. Now they are many more possibilities as the 6 can go in the ones column or the tens column. Again, you can list them on the board as they are suggested by the class.

Then, introduce the interactivity, explaining that now we are being given a digit by the computer and our challenge is to find the largest possible even number with another digit of our choice. Children could write the answer to the computer-generated problem on individual whiteboards, or they could discuss with a partner before offering their suggestion. If the computer does not agree with the class' suggestion, take some time to discuss why. You will find that the interactivity does not allow a zero to be used as the tens digit of a two-digit number. If this comes as a surprise to the class, encourage them to consider why.

Ideally at this point, each pair of learners would have access to the interactivity on a computer or tablet so they can have a go at some more examples. Invite them to develop a strategy for always finding the highest even number on their first attempt. If you do not have access to multiple devices, you could continue working as a whole class using the interactivity displayed, or you could ask pairs to use digit cards to model what the interactivity does. 

Bring everyone together after a suitable length of time to share ways of working. Encourage a few pairs to articulate how they make sure their first 'guess' is always the highest possible two-digit even number. You could compare and contrast the different approaches.

As a final challenge, give everyone time to consider how their strategy would change if they were asked to create the highest possible two-digit odd number. You can change the settings of the interactivity by clicking on the purple cog in the top right corner, and selecting 'Even False'. The observant amongst your class may notice that the interactivity now offers an extra button which says 'I don't think there is such a number'. When will this be useful? Why?

Key questions

What do you know about even numbers?
What do you know about the ones digit of even numbers?
Where could the digit the computer has given us go - in the ones or tens?  Why?
How can we make a large number?
How do we know that is the largest even number we can make?

Possible support

Some children might benefit from having equipment to help them check whether a number is even, for example multilink cubes or counters that can be put in pairs.

Possible extension

Change the settings on the interactivity (click on the purple cog) to challenge children to extend their strategy for three-digit and four-digit numbers. They can choose to be given one, two or three digits by the computer (but the number of digits provided must be fewer than the target).

The problem Dozens includes a similar interactivity but with a more challenging starting point and the scope to explore multiples of any number, not just odd/even numbers. You could take a look at the Teachers' Resources of Dozens to see how that interactivity might be used in the classroom.